Topology

As is tradition, the prize pool has increased (to $300 this year).

I have collapsed first and second place into a winner-takes-all arrangement (c’est la vie).

Furthermore, there are additional changes to the structure of this Game:

1. you must now pass the problem set to be awarded the prize money;
2. you may submit your solutions to the problem set at any point in the future;
3. if you plagiarise work, I reserve the right to ban you from all subsequent competitions — grim trigger
4. the problem and solution set will now be courteously supported by MathJaX, TikZ, and my own JavaScript
   • the problems can be found here, whilst the PDF can be found here and here (embedded).
   • my solutions will be available from the start of 2026; by viewing them you forfeit the prize money
5. Good luck!

PDF#

Links#

• Submission link
• List of Past Winners

Structure#

Most notably, the structure from this year has changed. Instead of just offering a single PDF and then writing up solutions on this site, the problems themselves are accessible from below and once 2025 transpires, my solutions will be available as toggled nested environments.

notes#

these are some more informal notes that I have made after having taken the rigorous Analysis course.

• \(c_{00}\) can be thought of as ‘finite vectors’. all \(\mathbb{R}^n\) tuples can be written as elements of \(c_{00}\) with infinitely many zeros after the $n$th term.
• \(c_0\) are all sequences that converge to zero. thus they will include all those in \(c_{00}\) and more.
• \(\ell^2\) are the vectors that fade fast enough for their energy 𐃏 to stay finite
• \(\ell^\infty\) are the infinite vectors that never blow up – their entries are bounded.

\[c_{00} \subset c_0 \subset l^\infty \]

I am finding Real Analysis to be more difficult than any other mathematics that I have studied before. I can seem to verify the truth of statements because they seem right; but I am having a difficult time producing rigorous and correct proofs.

It seems that High-School children (on the internet) are able to self-study Fomin with success. Bitterly, we remind ourselves:

“Comparison is the thief of Joy”—Theodore Roosevelt (probably)

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